The OpenSSH server, among others, requires your public key to be given to it in a one-line format before it will accept authentication with your private key. (SSH-1 servers also used this method.)
The ‘Public key for pasting into OpenSSH authorized_keys file’ gives the public-key data in the correct one-line format. Typically you will want to select the entire contents of the box using the mouse, press Ctrl+C to copy it to the clipboard, and then paste the data into a PuTTY session which is already connected to the server.
See section 8.3 for general instructions on configuring public-key authentication once you have generated a key.